Can limits‐to‐arbitrage from bounded storage improve commodity term‐structure modeling?

• DOI: http://doi.org/10.1002/fut.22006
• Date: 01 July 2019
• Author: Tore S. Kleppe,Atle Oglend
• Published date: 01 July 2019

Received: 8 March 2018

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Revised: 25 February 2019

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Accepted: 1 March 2019

DOI: 10.1002/fut.22006

RESEARCH ARTICLE

Can limits‐to‐arbitrage from bounded storage improve

commodity term‐structure modeling?

Tore S. Kleppe

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Atle Oglend

Department of Mathematics and Physics,

University of Stavanger, Stavanger,

Norway

Correspondence

Atle Oglend, Department of Industrial

Economics, University of Stavanger, 4036

Stavanger, Norway.

Email: atle.oglend@uis.no

Abstract

This paper develops and estimates a two‐factor competitive storage model for the

purpose of pricing commodity futures. The empirical relevance of the model is

evaluated for US natural gas and crude oil futures by comparing the pricing

performance to reduced form models. Results suggest jump models, both reduced

form and economic, improve modeling due to incorporating pricing discontinuities.

Furthermore, the economic model precludes carry arbitrage, which appears

relevant for pricing natural gas futures. For crude oil, the reduced form models

produce superior pricing under nonstationary market conditions, and the economic

model produces superior long‐dated futures pricing under stationarity.

KEYWORDS

commodities, futures, storage, term‐structure modeling

JEL CLASSIFICATION

C13, C15, D22

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INTRODUCTION

Commodities forward and futures are important assets allowing producers, consumers, and other commodity

stakeholders to hedge or gain exposure to commodity price risk and plan economic activity (Bhardwaj, Gorton, &

Rouwenhorst, 2015; Gorton & Rouwenhorst, 2006). The distinguishing feature of commodities as an asset class is the

convenience yield, often defined as the (net) flow of services that accrues to a holder of the physical commodity, but not

to a holder of a contract for future delivery (M. J. Brennan, 1991). As such, modeling the term structure of commodities

requires proper modeling of the convenience yield.

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With storage the commodity becomes a capital asset, and the convenience yield can be interpreted as an option premium

on the flexibility of storage in states of aggregate scarcity of marketable supply (Dockner, Eksi, & Rammerstorfer, 2015;

Heaney, 2002, 2006; Hochradl & Rammerstorfer, 2012; Milonas & Thomadakis, 1997a; Routledge, Duane, & Chester, 2000).

This model of the convenience yield is embedded in the competitive storage model of commodity price dynamics (Deaton &

Laroque, 1992; Scheinkman & Schechtman, 1983; Wright & Williams, 1991) through the shadow value of storage constraints

(or marketable supply). The option model of convenience yields has previously been applied to derive empirical measures of

the convenience yield (Dockner et al., 2015; Heaney, 2002, 2006; Hochradl & Rammerstorfer, 2012) and to demonstrate its

relevance for futures pricing (Basu & Mire, 2013; Heaney, 2006; Hochradl & Rammerstorfer, 2012).

Most of the literature on commodity term‐structure modeling is based on adaptations of affine term‐structure models

from finance. This literature can be traced to the seminal contribution of Black (1976). The standard approach is to directly

J Futures Markets. 2019;39:865–889. wileyonlinelibrary.com/journal/fut © 2019 Wiley Periodicals, Inc.

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The immediate commodity return under the risk‐neutral measure is , where is the short rate, is the instantaneous explicit cost of storage, and is the convenience yield. Since the volatility of the short

rate and explicit cost is in general orders of magnitude lower than the convenience yield, the major variation in the term structure is given by the convenience yield.

or indirectly define an exogenous affine jump diffusion (AJD) process (Duffie, Pan, & Singleton, 2000) for the convenience

yield (and possible other pricing factors). In the direct approach, the convenience yield is a primitive, and the term structure

is the resulting expected future spot price under the risk‐neutral measure (Casassus & Collin‐Dufresne, 2005; Cortazar &

Naranjo, 2006; Doran & Ronn, 2008; Gibson & Schwartz, 1990; Hilliard & Reis, 1999; P. Liu & Tang, 2011; Schwartz, 1997).

The indirect approach follows Heath, Jarrow, and Morton (1992) and defines the risk‐neutral dynamics of the term structure

itself as the primitive, whereby the convenience yield process can be backed out indirectly (Crosby, 2008; Miltersen &

Schwartz, 1998; Trolle & Schwartz, 2009). The AJD processes are convenient as they lead to flexible affine term‐structure

models that are fairly straightforward to estimate.

In this paper, we develop and estimate a two‐factor competitive storage model that endogenously produces a

convenience yield as the shadow value of storage constraints. Marketable supply from storage is bounded either at zero

or full capacity. When either constraint binds, there is no economic restriction that connects the spot price to prices for

future delivery, and the aggregate scarcity of either stocks (in a stock‐out state) or capacity (in a full capacity storage

state) creates an economic profit (scarcity rent) to those who own such assets. Empirically, this manifests as

discontinuous jumps in the slope of the term structure. Indeed, like the price of an asset being determined by the

discounted flow of dividends, the entire competitive price of the commodity can be shown to consist of the discounted

expected convenience yield flows (Oglend & Kleppe, 2017), which like this paper estimates the storage model under

completely bounded storage (zero and upper capacity bounds). The model is a standard competitive storage model,

except for the introduction of two fundamental shocks—a temporary and a persistent net‐supply shock. As we show

below, two shocks are necessary to allow sufficient flexibility in the term structure. Importantly, two shocks allow the

model to separate level shifts from slope shifts in the term structure.

The empirical pricing relevance of the economic model and its restrictions are evaluated by comparing its

performance to equivalent reduced form pricing models, specifically a linear VAR model with GARCH(1,1) errors, and

a maximal affine Gaussian two‐factor pricing model with and without a jump factor, as introduced by Casassus and

Collin‐Dufresne (2005). Jump dynamics has been shown to be relevant for commodities (Bjursell, Gentle, & Wang,

2015; Chan, Wang, & Yang, 2009; Koekebakker & Lien, 2004). The competitive storage model in this paper is essentially

a restricted two‐factor model with jumps in spot prices, and so is comparable to reduced form two‐factor jump models

for commodity pricing. Furthermore, to secure informative comparisons, all models are estimated using the same data,

and the reduced form and economic models are estimated consistently using the same methodology.

Futures pricing performance is evaluated for two empirical cases: US natural gas and US crude oil futures markets.

The analysis suggests that jump models, both reduced form and economic, improve term‐structure modeling due to

their ability to incorporate discontinuities in the term‐structure slope. Furthermore, the economic model precludes

carry (cash) arbitrage. This is not the case for the reduced form pricing models, as noted by Ribeiro and Hodges (2005).

For natural gas, precluding carry arbitrage appears relevant to produce an empirically relevant asymmetric basis as a

function of storage (working curve). Hochradl and Rammerstorfer (2012) shows that short‐sale constraints or restricted

storage access is informative on the convenience yield in European Natural Gas Hub trading.

The crude oil case shows that the reduced form models produce superior pricing under nonstationary market

conditions, while the economic model produces improved long‐dated futures pricing in a stationary market. The

economic restrictions allow unique identification of underlying shocks, and subsequent state variables, when using only

the spot and slope of the term structure. The reduced form models have no a priori restrictions on factor persistence,

which makes identification of implied price and convenience yield processes more difficult with only spot and slope

information. The results suggest that estimation of reduced form factor models without persistence restrictions needs to

include information on as much of the term structure as possible to identify long‐run level from slope components.

Furthermore, to be more practical for real‐world nonstationary markets, the storage model should be extended to include

stochastic long‐run prices.This point is also mentioned by Legrand (2018) in his survey of the competitive storage model.

To apply our model for pricing purposes, we need to solve numerically for the rational expectations pricing function

that defines the mapping from underlying shocks, or factors, to the competitive equilibrium price. Once the pricing

function is found, the shocks are identified using observations on spot prices and the nearest part of the term structure

—a similar procedure of state identification as in Pirrong (2011). The identified shocks allow us to model the term

structure using a nonlinear VAR approximation to the rational expectation equilibrium. Approximating the equilibrium

by a nonlinear VAR model is a novelty of this paper. We choose this approach as it opens up for well‐established tools

for estimation and analysis of VAR models, and for easier application of the model for pricing purposes. The economic

and factor models are estimated using Bayesian inference methodology. This falls into the Sims tradition of VAR

approximations to dynamic stochastic general equilibrium models. Using derivative prices to allow more complicated

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